The ratio of the radii of two cones is 1:2 and the ratio of their heigths is 2:1. Find the ratio of their volumes.

Answer

To find the ratio of the volumes of two cones given the ratios of their radii and heights, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Ratios**:
- The ratio of the radii of the two cones is given as \(1:2\).
- The ratio of the heights of the two cones is given as \(2:1\).

2. **Assign Variables**:
- Let the radius of the first cone be \(r_1 = x\) (where \(x\) is a variable).
- Then, the radius of the second cone will be \(r_2 = 2x\) (since the ratio is \(1:2\)).
- Let the height of the first cone be \(h_1 = 2y\) (where \(y\) is another variable).
- Then, the height of the second cone will be \(h_2 = y\) (since the ratio is \(2:1\)).

3. **Volume Formula for a Cone**:
- The volume \(V\) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]

4. **Calculate the Volume of the First Cone**:
- Using the values for the first cone:
\[ V_1 = \frac{1}{3} \pi (r_1^2) (h_1) = \frac{1}{3} \pi (x^2) (2y) = \frac{2}{3} \pi x^2 y \]

5. **Calculate the Volume of the Second Cone**:
- Using the values for the second cone:
\[ V_2 = \frac{1}{3} \pi (r_2^2) (h_2) = \frac{1}{3} \pi ((2x)^2) (y) = \frac{1}{3} \pi (4x^2) (y) = \frac{4}{3} \pi x^2 y \]

6. **Find the Ratio of the Volumes**:
- Now we find the ratio of the volumes \(V_1\) to \(V_2\):
\[ \text{Ratio} = \frac{V_1}{V_2} = \frac{\frac{2}{3} \pi x^2 y}{\frac{4}{3} \pi x^2 y} \]
- Simplifying this gives:
\[ \text{Ratio} = \frac{2}{4} = \frac{1}{2} \]

7. **Final Result**:
- Therefore, the ratio of the volumes of the two cones is \(1:2\).